Backward uniqueness of semigroups arising in coupled partial differential equations systems of structural acoustics. (English) Zbl 1109.35065

The problem of backward uniqueness is considered for two structural acoustic models: one of them couples a hyperbolic equation with an elastic plate equation (parabolic type) defined on its elastic wall, and the another couples the same equation with a thermoelastic plate equation (parabolic or hyperbolic-dominated type) defined on its flexible wall. These structural acoustic models read as systems of PDE are proved to generate the first order abstract Cauchy problem with the generator \(A\) being neither the generator of an analytic semigroup nor of a group (these cases were studied earlier). In the paper the operator \(A\) is proved to be the generator of a strongly continuous semigroup and special estimates for the resolvent of \(A\) are shown to hold in a Banach space. Using results for such semigroups the backward uniqueness theorems for the models considered are obtained.


35L20 Initial-boundary value problems for second-order hyperbolic equations
35R35 Free boundary problems for PDEs
76Q05 Hydro- and aero-acoustics
47D06 One-parameter semigroups and linear evolution equations
35L90 Abstract hyperbolic equations